
which indicates that \(\left({\mathbf {\beta }}_n^{(\text{c})}\right)^H {\hat{\mathbf {h}}}_n^{(\text{c})}\)  equals to the norm of the projection of \({\hat{\mathbf {h}}}_n^{(\text{c})}\)  on \(\mathbf {G}_1\) . Obviously, \({\hat{\mathbf {h}}}_n^{(\text{c})}\)  is independent of each column of \(\mathbf {G}_1\) . Hence, each element of \(\mathbf {G}_1^H {\hat{\mathbf {h}}}_n^{(\text{c})}\)  follows i.i.d. \({\cal {CN}}(0,\delta _n^{(\text{c})})\) . According to [1]}, \(\frac{1}{\delta _n^{(\text{c})}} \left\Vert \mathbf {G}_1^H {\hat{\mathbf {h}}}_n^{(\text{c})}\right\Vert _2^2 \sim {\cal W}_1 (B- b_\text{c} - b_\text{d},1)\)  is a central complex Wishart random variable with \(B- b_\text{c} - b_\text{d}\)  degrees of freedom. Then, we have
\({\mathbb {E}} \left\lbrace  \frac{1}{S_n^{(\text{c})}}\right\rbrace =\frac{1}{q_{{\text{s}},n} u_n^{(\text{c})} (B- b_\text{c} - b_\text{d} -1)\delta _n^{(\text{c})}}.\) 
